The metric introduced in your question is not described in the commonly known publications related to the metric spaces. In order to verify your idea I would suggest to consult the Encyclopedia of Distances

http://www.springer.com/gp/book/9783662518687

It is not clear what are the covariant base vectors in the proposed metric?

(The covariant metric tensor is defined as g_{\mu\nu}=e_{\mu} \dot e_{nu})

If we can resolve this issue the additional questions are easy to answer using the basic formula of differential geometry.

As far as the circular metrics are concerned, the most popular is a metric proposed by K. Gödel; an interesting study describing properties of the world with such a metric is presented at

http://iopscience.iop.org/article/10.1088/1367-2630/15/1/013063

The technique of ray tracing described in the above publication is very useful to investigate any new and exotic metric spaces.

If modifying the dy, and dz, it will be a (half) Helix, and the Lorentz metric will depend of dz and vt (into an axial direction).

Hi Janusz,

what is exactly unclear concerning the covariant base vectors in the proposed metric?

For dx,dy,dz

Hi Wolfgang,

Thank you for your comments. My question concerning the base vectors was inspired by the lack of an explicit definition of the coordinates. It is not obvious how you can cyclically map the Cartesian coordinates in R^3 to circles; the explicit form of this mapping will be very helpful.

After the mapping is defined there is no problem in calculating the Ricci tensor. Most of the calculations could be either performed or verified using the Wolfram Mathematica.

Kind regards,

Janusz

Hi Janusz,

The mapping is very simple.

x,y,z are the coordinates in the considered space.

X,Y,Z  are the coordinate values mapped to circles with radius R with 0

Hi Wolfgang,

Thank you for the explanation. With this definition the Universe is a three-dimensional torus. The cosmological model with such a geometry was proposed in 1984 by Starobinski and Zeldovich.

The topology of a 3-D torus is discussed on page 22 in

https://arxiv.org/pdf/gr-qc/9605010.pdf

where we can find very pertinent mathematical derivations. In particular the Ricci tensor and the scalar curvature are given in eq. (7).

I hope that this answers your question at least in part.

Kind regards,

Janusz

An artist's point of view 

https://www.google.se/search?q=reutersv%C3%A4rd&client=ms-android-sonymobile&prmd=imnv&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjV15GvhuHUAhXmF5oKHS97CIIQ_AUICSgB&biw=360&bih=578

Hi Janusz,

The document indeed answers nearly all aspects of my question.

The last sentence in the conclusion  "Thus, at least from a theoretical point of view, the field of cosmic topology appears not closed but, on the contrary, in a promising state of development" confirms that there are still many open questions in cosmology. 

The remaining part of my question for which in a first glance I could not find an answer in the document is:

Does a constant non zero curvature in space  cause a frequency shift ?

Kind regards,

Wolfgang

Hi Wolfgang,

The last part of your question is the most difficult because according to the observations the redshift exhibits intriguing cyclic behavior. Some conjectures suggest explaining this fact by the three torus topology defined in your question. The entire problem is discussed in

http://adsabs.harvard.edu/full/1990A%26A...239...24F

I hope that you will be able to answer the remaining questions after completing your project.

On the very technical level (returning to the first part of your question) the specific formula for the metric tensor and for the Ricci tensor on a two torus are derived at

http://www.rdrop.com/~half/math/torus/torus.geodesics.pdf

The same relations can be obtained for a three torus.

Plotting of a three torus is discussed in the Mathematica stack exchange

https://mathematica.stackexchange.com/questions/23546/how-can-i-draw-a-3d-cross-section-of-a-3-torus-embedded-in-4d-euclidean-space

Kind regards,

Janusz

Hi Janusz,

thank you again for the verey comprehensive answer.

May be the difficulties in the question about a constant curvature causing a frequency shift can be reduced to the following question:

"Do parallel geodesics with non zero distance  exist in a space with non zero  constant curvature". (On the earth surface at least one of two parallel circles is no great circle.) (parallel means with constant minimum distance along the path)

If they don't  exist, the following would happen to a photon, which moves in the space:

The spatial extension of the photon perpendicular to the movement is given by the varying minimum distance of neighboured geodesics. A resulting cross section modification will surely  have an influence on the wavelength.

Hi Wolfgang,

Based on my preliminary analysis the geodesics with nonzero distances are not parallel. The mechanism of frequency shift which you propose should be therefore considered seriously. The entire problem still requires a systematic analysis.

The work with charting of the geodesics on a three torus is in many ways similar to the problems of trajectory analysis in the dynamical systems.

The influence of the topology of a three torus on frequency shift is also discussed in the thesis

http://www.diva-portal.org/smash/get/diva2:754277/FULLTEXT01.pdf

Kind regards,

Janusz

Hi Janusz,

Thank you for the again very comprehensive answer. I am also convinced that a serious analysis is required, which considers parallel transport in the curved space and takes into account the (not precisely known) spatial extension of photons.

Kind regards,

Wolfgang

Hi Janusz

What do you think about the following geometry:

At each point, every straight line( or geodesic line) is topologically equivalent to a surrounding bundle of tangent circles with a giant radius. Any infinitesimal small displacement in this geometry is accompanied with a random process selecting a specific one out of that 360 degree bundle of circles. The displacement then takes place on that specific circle.

It is not even clear that this kind of space is a semi Riemannian manifold. Intuitively even a single geodesic has diverging properties. Something like parallel transport seems to be undefined.

Sincerely ,

Wolfgang

Hi Wolfgang,

Thank you for the interesting question.

My first impression could be succinctly described by the quote from E. T. Bell: "The longer mathematics lives the more abstract - and therefore possibly also the more practical - it becomes"

I will try to analyze the properties of this geometry formally. Looking at the complexity of the available astrophysical data I expect that the proposed by you random process is the necessary additional "degree of freedom" .

As far as the problem of the parallel transport is concerned it will be defined in the different manner in the proposed geometry.

Sincerely,

Janusz

Hi Janusz

Thank you for the very positive comment.

An interesting consequence of the geometry is that it would allow explaining red shift and extended time scale of supernova explosions very simple:

Due to the curved path, light needs more time arriving from a very distant point compared with a virtual path along the Euclidian distance to that point.

Sincerely,

Wolfgang

Hi Wolfgang,

Thank you for the stimulating discussion. I think that the topology of space can indeed explain the "red shift" in much more natural way than other theories. It is reassuring that you are still searching for the "true" geometry of the surrounding space.

The general ideas of Riemann outlined in his seminal paper

http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf

are still very actual (despite that some of the manifolds which will be proposed in the future are not necessarily Riemannian).

Sincerely,

Janusz

Dear Wolfgang,

The debate on topology of our Universe is still ongoing and all possibilities are open. You can find a wealth of the experimental facts in the following link

http://home.fnal.gov/~kubik/topo_cosmo_dk11.pptx

It is particularly impressive how ingenious are techniques used to evaluate the red shift. I'm sure that this presentation will be useful for you.

Sincerely

Janusz

Dear Janusz,

the PowerPoint presentation indeed impressively illustrates many  cosmologic studies.

A very basic point is the question, if a finite Universe automatically implies a geometrical red shift. The argumentation could be as follows:

A light beam establishes a communication line between two points A and B. If we extend the separation of the points, we reach a distance where the geodesic connection is significantly curved. Because it is curved, a signal from A to  B needs more time as on a virtual straight connection. That means that the effective signal speed or analogously the signal transfer rate from A to B now is reduced. As a consequence, the output transfer rate of that very long communication line is smaller as the input transfer rate. This is exactly what would cause a red shift.

Do you think this simple argumentation is substantial, or is there a gap in the arguments?  

Sincerely

Wolfgang

Dear Wolfgang,

I agree with you in principle but I would like to suggest some changes to the wording used in your argumentation. I have to work on this a little. In the mean time please have a look at the text which was posted on RG by J. G. Von Brzeski

(The link is very long but it works)

https://www.researchgate.net/profile/J_Von_Brzeski/publication/228682270_Expansion_of_the_Universe-Mistake_of_Edwin_Hubble_Cosmological_Redshift_and_Related_Electromagnetic_Phenomena_in_Static_Lobachevskian_Hyperbolic/links/02e7e52a0a5d84a843000000/Expansion-of-the-Universe-Mistake-of-Edwin-Hubble-Cosmological-Redshift-and-Related-Electromagnetic-Phenomena-in-Static-Lobachevskian-Hyperbolic.pdf?_iepl%5BhomeFeedViewId%5D=Ht9AUrhEzhtgAchnQA7kK6OpxMOMokQmra4Z&_iepl%5Bcontexts%5D%5B0%5D=pcfhf&_iepl%5BinteractionType%5D=publicationDownload&origin=publication_detail&ev=pub_int_prw_xdl&msrp=mtXPBudPStO-pXCZt30zx1oMd8eeWHsdJ_1C_MmKpNhzyuU1_lj820bLmv_5_VLD64nj_bQNgbsa9RgKEbmlaloy3G2SBStnZDFgxl8ItTP9VnvfeCQ2iiM2.IY1JE2RO0ApF2M3o5j1md5bVhdVlIbkB3bGTo9a8cH4GVHUjRA5j4oP8k-26q96BJq2Wa1R2rZC0aTSJPWuBSToB0_j4u6j2xZgSvA.rQzmAATJWOtGINKX2JSBt2Hc6fjRkHLPWCRKTgM6CyuFemcRm2dm6Vi40AI2al2rI4cQ0W46K6ZLYgQfIp2UstOIGLf8CQcQ0lFYZw.YuWyk8Qi770BXW8fqyGD0VyWxBDwO3g9HdaqKQDhzpcijaRHXXliNuulgnV50EP07Y9FbKbShQHndn9LozqC59ca2nwaxx4uUEw7Tw

The concept of Lobatchevsky-Hubble shift introduced above is quite useful and interesting. My suggestion in essence will follow this reasoning (but not exactly). I will write more on this subject tomorrow.

Sincerely,

Janusz

Article Expansion of the Universe-Mistake of Edwin Hubble? Cosmologi...

Hello Wolfgang,

After extensive reading of the additional papers and following your "parallel question" on the research gate I do not see yet any convincing answer. This is not surprising considering the complexity of the problem. I would like to suggest solving the covariant form of the Maxwell equations assuming certain form of the metric tensor. This approach will contain all effects of the curvature of space including the T^3 topology as well as the gravity related perturbations due to the given mass distribution. In this approach we can include also the effect of attenuation of electromagnetic waves by the cosmic dust (this effect is often neglected without any justification).

Perhaps you are aware about such a study?

Kind regards,

Janusz

Hello Janusz,

Currently I am focusing on pure geometry and will not analyse possible physical consequences without a promising concept.

But what do you think about the following geometry:

Given two cartesian spaces G and U with three and six dimensions and a finite length R.

X(x,y,z) is a point in G with the coordinates x,y,z.

 A(a,u,b,v,c,w) is a point in U with the coordinates a,u,b,v,c,w

Also given is  a mapping u: G-->U

A = u(X)

a=R*sin(x/R), u=R*cos(x/R),

b=R*sin(y/R), v=R*cos(y/R),

c=R*sin(z/R), w=R*cos(z/R)

and the inverse mapping g: U-->G

X=g(A)

x=R*atan2(u,a)

y=R*atan2(v,b)

z=R*atan2(w,c)

also given is a metric D in U.

D(A1,A2)²=(a2-a1)²+(u2-u1)²+(b2-b1)²+(v2-v1)²+(c2-c1)²+(w2-w1)²

and a (non Euclidian) metric d in G, based upon D:

d(X1,X2)= D(A1,A2) with A1=u(X1), A2=u(X2);

The unity vector of a direction in point X1 towards point X2 is given as the direction towards an inverse mapping of an infinitesimal shifted point A1’ (with parameter e) towards the mapping A2 of X2 in U.

A1=u(X1); A2=u(X2), A1’= A1+e(A2-A1);

E{(g(A1’)-X1)/e} with E{X} =X/|X|

With the above properties, G is finite, homogeneous and isotropic.

If a position X2 is the result of a displacement X1 + DX, then X2 is given as

X2=g(u(X1+DX)). This mapping ensures the finity of G and causes that G is nested in all directions.

The physical consequence  of this geometry could be:

The difference between a Cartesian metric c in G and the given metric d leads to a red shift proportional to d/c;

The modified metric and the modified direction relation have an influence on gravitational forces. (May be this makes dark matter obsolete)

Sincerely,

Wolfgang Konle

Hello Wolfgang,

The metric described in your comment is essentially analogous to the metric defined in your original question. I will analyze solution of the covariant form of the Maxwell equations with this metric to resolve the issue of the red shift.

I will let you know what is my conclusion in few days.

Kind regards,

Janusz

Hello Janusz

There is also an impact on the direction and on the value of gravitation. Do you know any work with a similar approach.

Kind regards,

Wolfgang

Hello Wolfgang,

Thank you for the information. I'm not aware about your approach being used in any other studies; it looks original. Currently I'm exploring the covariant form of the Maxwell equations in different metrics. The main part of this work focused on writing a program in Mathematica. I hope to get some results this week.

Kind regards,

Janusz

Hi Janusz,

I also had a quick look on the possible consequences on the covariant Maxwell equations. Considering all dimensions and full nonlinearity looks like a tremendous effort. Fortunately the geometry has various symmetry properties (constant curvature is one property)  and there is also the possibility to focus on the difference to the standard metric (applying a kind of perturbation calculation)

May be focusing on an axial symmetric case makes it is easier to avoid getting lost in a large number of mathematical expressions whose interpretation could be difficult.

Kind regards,

Wolfgang

Hi Janusz,

the maths show, that the proposed homogeneous, isotropic and closed space is locally flat.

The curvature is exactly zero. But this is not the end of the story, because the Ricci tensor only provides a local view. Therefore, solving the Maxwell equations and taking into account (local) space curvature does not lead to another result as in an Euclidian space. But the following view about non parallel diverging geodesics leads to something else:

Consider a 6 dimensional space defined as follows:

The original coordinates are:  x, y, z, the new coordinates are: a, u, b, v, c, w.

The coordinate transformation is: a=R*sin(x/R), u=R*cos(x/R), b=R*sin(y/R), v=R*cos(y/R), c=R*sin(z/R), w=R*cos(z/R)

and inversely

x=R*atan2(a,u), y=R*atan2(b,v), z=R*atan2(c,w).

In this space, we consider a photon starting at Point (x0,y0,z0) in direction of z. Then we observe the point (x0+dx,y0,z0) and recognize, that at this point, the z-direction is turned in y-direction by the extremely tiny angle dx/R in respect to the z-direction at Point (x0,y0,z0).

Observing the point (x0,y0+dy,z0) accordingly shows a z-direction altered  in  x-direction by dy/R. (A displacement dz does not alter the z-direction.)

[It is not even necessary to calculate the geodetics direction dependency on the displacement, because it is a direct consequence of the isotropy] 

The altered neighbored z-directions (which are geodesics) mean, that on its way in z-direction the photon experiences the area perpendicular to its pointing vector diverging with a rate proportional to (z-z0)dx*dy/R². This means that the energy density of the photon becomes reduced with a corresponding tiny rate.

This is not a proof, but a very plausible scenario.

Hello Wolfgang,

Thank you for the information. Your statement about the Maxwell equations in this metric is correct.

As far as the effect of divergence of geodesics is concerned I agree with your interpretation. The proposed model is also consistent with the "red shift effect" in the General relativity (for photon moving in the gravity field).

In general I consider your theory of topologically induced red shift to be quite elegant, and for all of us who are in favor of the Occam's principle, worth of a serious consideration. There is still a lot of work to formalize it and to perform the experimental evaluation.

In the recent years the experiments were performed to measure a transverse quantum state of light in the context of the quantum entanglement

https://arxiv.org/pdf/quant-ph/0507142.pdf

Perhaps the equivalent models could be designed to support your theory?

Kind regards,

Janusz

Hello Janusz,

thank you for your encouraging answer and the specific information.

As a next step I will try analyzing line integrals along neighboured geodesic circles in the proposed space. The aim is to find out, what happens to a circular area perpendicular to a single geodesic during travelling along the geodesic.

If we can prove an expansion of this perpendicular area, we have found an expanding space in a stationary geometry.

Kind regards,

Wolfgang

Hello Wolfgang,

Thank you for your letter and the update. I expect that your conjecture could be proven leading to the very interesting result. Some of these calculations are scattered in the literature, I will let you know after completing my survey.

Kind regards,

Janusz

Hi Janusz,

The six dimensional geometry I had proposed up to now only describes the closed nature of the space, but does not include the topology.The complete model now connects any displacement ds in the Euclidian space with a rotation angle ds/R in an also three dimensional image space.

Proofing dilatation effects along diverging geodesics is then reduced to check the result of two subsequent rotations. First rotate with a tiny angle dx/R and then with a large angle delta_z/R. Finally compare the result with a single rotation delta_z/R. The resulting positional difference is not dx as it would be in a proper Euclidian space after subsequent displacements.

The first rotation has an extremely tiny effect on the direction of the rotation axes for the second rotation. But this tiny effect causes the divergence of the geodesics.

I try to calculate this effect quantitavely.

Kind regards,

Wolfgang

Hello Wolfgang,

Thank you for the update. The superposition of two rotation is indeed the most elegant approach; you can apply also the quaternion formalism.

In order to connect your work with other approaches used to study the red-shift I can recommend the thesis

http://www.diva-portal.org/smash/get/diva2:754277/FULLTEXT01.pdf

Even after reading the abstract I see that the objectives of the Author are consistent with your ideas (this observation applies of course to the general directions of the research only). The main point is that the cosmological red shift can be explained using the topological reasoning.

Kind regards,

Janusz

Hi Janusz

Thank you for the text of this thesis with with a very detailed and comprehensive mathematical analysis of Bianchi type I space times. 

With an impressive set of mathematical  methods, this thesis provides different consistent models for a red shift in an expanding universe. All provided methods are based on the (local) differential geometrical properties of the space. Topological properties like long distance effects of diverging geodesics have not been considered.

This thesis shows, that all explanations  of the red shift, based on differential geometric properties (which only consider the local properties of the space) need space expansion for explaining the red shift.

It seems that mathematicians tend to ignore the fact, that the concept of Riemannian curvature is based on parallel transport over a small closed path.

Hello Wolfgang,

Thank you for your comments. I wonder if the effects of the diverging geodesics could be combined with the local effects? In the Universe you proposed I expect to have the matter and the red shift caused by the GR effects (in the static universe sense) will be superposed with the redshift conjectured in your theory.

Are you planning to address this problem?

All the best for your research with the fundamental problems of space topology.

Janusz

Hi Janusz

my personal feeling is that there are no (noticeable) local effects resulting from diverging geodesics with such a small divergency as resulting from the proposed model. (The Riemannian curvature resulting from the metric in the proposed geometry is zero.)

As another large scale effect (in addition to the red shift) I would expect an impact on gravitational forces between clusters of galaxies.

The focus of my future activities is getting  more widespread attention for that theory as a possible alternative to the big bang theory. May be you could contribute to an increasing attention.

Hello Wolfgang,

Thank you for the information. There are so many competing theories of the redshift and it is very likely that we have to consider many factors in any attempt to explain this phenomenon. Review of the existing explanations presented for example in

https://arxiv.org/vc/arxiv/papers/0806/0806.4085v1.pdf

confirms that we are still not sure what is the actual reason of the redshift. New theories are being proposed frequently such as the model of Wetterich (2013)

https://arxiv.org/abs/1303.6878

but there is no consensus. This observation justifies your efforts, I'm definitely interested in following this topic and I will try to write a short document summarizing our discussion and some recent findings.

Hi Janusz,

Thank you for the publications, which discuss alternatives to the big bang theory. Gray and Dunning-Davies consider a large variety of competing theories but do not take into account topological long distance effects of a curved geometry. Wetterich even implies relations between a cosmological scale and an atomic scale and discusses the possibility that basic physical constants on both scales are synchonously varying.

Do you think that up to now somebody has deeply analysed long distance effects of a curved space geometry? Did all scientists stop their "Red Shift Analysis" within possible geometries after the finding of zero Riemannian curvature? Is a geometry with homology of displacement and cosmic rotation too simple, to be worth a deeper analysis?

May be those qestions are helpful to increase attention on that homology theory.